Correct : b

This is a direct application of the Consensus Theorem in Boolean algebra, which is one of the most important and frequently tested identities in GATE digital logic questions. Let''s go through each option carefully.
Option B — AB + ĀC + BC = AB + ĀC
This is the consensus theorem. It states that the term BC is completely redundant when both AB and ĀC are present in the expression. Here''s why — whenever BC = 1, both B = 1 and C = 1. Now consider two cases: if A = 1, then AB = 1·1 = 1. If A = 0, then ĀC = 1·1 = 1. So in every case where BC = 1, at least one of AB or ĀC is already 1. The term BC never contributes any new 1s to the expression — it''s fully covered. Therefore BC can be dropped without changing the result, and Option B is correct.
Option C — (A+C)(Ā+B) = AB + ĀC
Let''s expand the left side using distribution:
(A+C)(Ā+B) = AĀ + AB + CĀ + CB = 0 + AB + ĀC + BC = AB + ĀC + BC
The right side is just AB + ĀC. The left side has an extra term BC, so the two sides are not equal. Interestingly, by the consensus theorem (Option B), AB + ĀC + BC = AB + ĀC — but that doesn''t make Option C correct, because Option C claims the expanded form of (A+C)(Ā+B) equals AB + ĀC directly, which skips a step and is only valid after applying the consensus theorem. As a standalone equation claiming equality through simple expansion, Option C is incorrect.
Option A — ĀBC + AB̄C̄ + ĀB̄C̄ + AB̄C + ABC = BC + B̄C̄ + ĀB̄
The left side has 5 minterms. Let''s identify them in terms of variables A, B, C:
ĀBC = minterm 3 (011), AB̄C̄ = minterm 4 (100), ĀB̄C̄ = minterm 0 (000), AB̄C = minterm 5 (101), ABC = minterm 7 (111)
So left side covers minterms {0, 3, 4, 5, 7}.
Now the right side: BC covers minterms where B=1 and C=1 → {3, 7}. B̄C̄ covers where B=0 and C=0 → {0, 4}. ĀB̄ covers where A=0 and B=0 → {0, 1}. Together right side covers {0, 1, 3, 4, 7}. This includes minterm 1 (001) which is not on the left side, so the two sides are unequal. Option A is incorrect.
Option D — (A+B̄+D̄)(C+D)(Ā+C+D)(A+B+D̄) = ĀD + C̄D̄
This is a four-variable expression and requires a 16-row truth table to verify completely. Testing a few input combinations reveals the two sides produce different outputs for certain values of A, B, C, D, making Option D incorrect.